If coupling between traces on the module and traces on the board being laid out is a concern, I think I'd leave it at the solder mask. Pretty much everything has a higher dielectric constant than air and the spacing is going to be smaller than pretty much any layers on your board, perhaps 5 mil (0.127 mm). With air in the gap you will see a lower capacitance.
The insulation is only an issue if the boards touch. I think that should be pretty hard to have happen unless the boards are being flexed which is inherently bad on its own. That will pop parts off the boards or damage them.
Skip the HASL and use ENIG. It doesn't really cost much more. I think Oshpark even offers it. Then there is no reason to worry about the vias.
The boards do touch if he SMT them. And I don't understand where the air gap is.
I don't think cross talk between traces is his concern, and if you look at RaspPi Pico, there are not many traces underneath, he can pour a ground plane under that area. My understanding is that he's worried about electrical isolation.
Maybe we mean different things by "touch". Of course they touch at the solder pads, but there will be a layer of solder between the boards. The data sheet even talks about through hole pins not reaching through the board, but there may be a blob of solder on the bottom side, so they recommend a pad underneath so the solder is not too thick and holds the module too far off the board. So you can count on a few mil of solder separating the two boards. I think he is planning to hand solder, so he may want to solder wick these pads to remove any blobs, even small ones. But he needs to be careful to not remove solder from within the through hole.
He may or may not be concerned with coupling between traces on the two boards, but that is not the same thing as saying there is or isn't a problem. A ground plane is clearly not a good idea for either insolation or for signal coupling. Best is to leave that layer a "keep out" zone, but I think he knows that, that's why he is worried about shorts, he plans to have traces and even vias under the module.
It might be best to do what I suggested, insert a spacer such as a corner of a piece of paper under the part to provide space until the corner pins are soldered. The space can be controlled by the thickness of the paper or the number of sheets. Remove them once the corners are tacked and then finish the rest of the pads. That will give a significant spacing between the boards, enough to prevent touching even if the boards are flexed.